Abstract
Routh reduction presents the minimum number of differential equations that uniquely describe the state of nonlinear mechanical systems where the state variables can be separated into essential ones and cyclic ones. This work extends Routh reducibility for a relevant set of controlled mechanical systems. A chain of theorems is presented for identifying the conditions when reduced order rank conditions can be applied for determining the Kalman controllability of Routh reducible mechanical systems where actuation takes place along the cyclic coordinates only, while some of the essential coordinates and their derivatives are observed. Four mechanical examples represent the advantages of using reduced rank conditions to check and/or to exclude linear controllability in such systems.
Highlights
Routh introduced his technique [22] for conservative mechanical systems in the form of a hybrid Lagrangian and Hamiltonian description
The Kalman controllability of cyclic mechanical systems is analyzed where external actuation is restricted to the cyclic coordinates, while the essential coordinates serve as the output states
= 0, Theorems 2, 5 and 6 imply that the Routh reduction cannot be carried out, and the equations of motion cannot be reduced to the essential coordinates only, even if the control torque is applied at the cyclic coordinate only
Summary
Routh introduced his technique [22] for conservative mechanical systems in the form of a hybrid Lagrangian and Hamiltonian description The advantage of this Routhian formalism becomes apparent when so-called cyclic coordinates are present in the system. The stability of certain steady-state motions can be analyzed by means of Lyapunov functions that are based on the so-called Routh potential [23]. To this day, the Routh reduction is still part of active research. The Kalman controllability of cyclic mechanical systems is analyzed where external actuation is restricted to the cyclic coordinates, while the essential coordinates serve as the output states The last example of a rotor model [10] represents the limitation of the reduction methodology when the Kalman controllability condition applied for the full state model cannot be simplified to reduced rank conditions
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